President-elect Donald Trump suggested he would be open to lifting sanctions on Russia and wasn’t committed to a longstanding agreement with China over Taiwan—two signs that he would use any available leverage to realign the U.S.’s relationship with its two biggest global strategic rivals.

In an hourlong interview, Mr. Trump said that, “at least for a period of time,” he would keep intact sanctions against Russia imposed by the Obama administration in late December in response to Moscow’s alleged cyberattacks to influence November’s election. But he suggested he might do away with those penalties if Russia proved helpful in battling terrorists and reaching other goals important to the U.S.

“If you get along and if Russia is really helping us, why would anybody have sanctions if somebody’s doing some really great things?” he said.

He also said he wouldn’t commit to America’s agreement with China that Taiwan wasn’t to be recognized diplomatically, a policy known as “One China,” until he saw what he considered progress from Beijing in its currency and trade practices.

THE TRUMP TRANSITION

The desire to change relations with Moscow in particular has been a goal of American presidents since tensions began rising under President Vladimir Putin’s leadership. Former Secretary of State Hillary Clinton sought the same goal early in the Obama administration, as did President George W. Bush, who met Mr. Putin early in his first term.

But Mr. Trump’s diplomatic efforts will have to compete with those in Congress, including many Republicans, who want to see the administration take a tough line with Russia after U.S. intelligence concluded that the government of Mr. Putin sought to influence the November presidential election with a campaign of cyberhacking.

Additionally, an unsubstantiated dossier of political opposition research suggesting ties between Mr. Trump and Russia was published this past week—drawing condemnation from Mr. Trump and his team but keeping Russian espionage in the spotlight. The allegations haven’t been validated by the U.S. intelligence agencies.

Mr. Trump in the interview suggested he might do away with the Obama administration’s Russian sanctions, and he said he is prepared to meet with Mr. Putin some time after he is sworn in.

“I understand that they would like to meet, and that’s absolutely fine with me,” he said.

Asked if he supported the One China policy on Taiwan, Mr. Trump said: “Everything is under negotiation including One China.”

China has considered Taiwan a breakaway province since Chiang Kai-shek’s Nationalists set up a government there in 1949, after years of civil war. Washington’s agreement to rescind diplomatic recognition of the government in Taiwan and uphold a One China policy was a precondition for the re-establishment of diplomatic relations between U.S. and China in 1979. Any suggestion in the past that the U.S. may change its stance has been met with alarm in Beijing.

It added, “we urge relevant parties in the U.S. to fully recognize the high sensitivity of the Taiwan question, approach Taiwan-related issues with prudence and honor the commitment made by all previous U.S. administrations.”

Though he has long been critical of China, Mr. Trump on Friday also made a point of showing a holiday greeting card he received from China’s leader, Xi Jinping.

President-elect Donald Trump made many promises in his first post-election news conference. Here is a look at the proposed timing for some of his pledges: lucas jackson/Reuters

“I have a beautiful card from the chairman,” he said.

Mr. Trump seemed impatient with diplomatic protocols involving China and Taiwan. After his victory he took a congratulatory phone call from Taiwan’s leader, triggering objections from Beijing and stoking concerns among some U.S. foreign policy experts who questioned whether he understood the implications of such a conversation.

Speaking of Taiwan, he said: “We sold them $2 billion of military equipment last year. We can sell them $2 billion of the latest and greatest military equipment but we’re not allowed to accept a phone call. First of all it would have been very rude not to accept the phone call.”

Mr. Trump has said in the past he would label China a currency manipulator after he takes office. In the interview, he said he wouldn’t take that step on his first day in the White House. “I would talk to them first,” he said.

He added: “Certainly they are manipulators. But I’m not looking to do that.”

But he made plain his displeasure with China’s currency practices. “Instead of saying, ‘We’re devaluating our currency,’ they say, ‘Oh, our currency is dropping.’ It’s not dropping. They’re doing it on purpose.

“Our companies can’t compete with them now because our currency is strong and it’s killing us.”

The interview came at the end of the week in which Mr. Trump saw much of his national-security team get closer to their appointments but had to push back against the Russia allegations and against criticism from ethics experts of his plan to maintain ownership of his business interests.

Six of his cabinet choices had confirmation hearings, and a number look likely to sail through. Many Democrats offered eager support for his pick for defense secretary, retired Gen. James Mattis.

Mr. Trump also brought his son-in-law, Jared Kushner, on as a senior White House adviser, although the appointment could be challenged under antinepotism laws. And he got closer to fulfilling a campaign promise as the Senate and then the House took procedural steps that begin rolling back or repealing the Affordable Care Act.

“He got elected as a fighter and he’s going to be president as a fighter,” said Ed Brookover, a former Trump campaign adviser. He added that Mr. Trump “is going to be a very active president and push a lot of buttons along the way.”

At a jam-packed news conference on Wednesday morning, Mr. Trump was both combative and flattering, shouting down one journalist but praising news outlets who he said covered him fairly. During the session, he accused intelligence agencies of allowing the dossier information to be leaked, and on Twitter he said they were employing the tactics of Nazi Germany. James Clapper, the director of national intelligence, said he doesn’t believe intelligence officials leaked the information.

Amid a flurry of questions about the dossier, Mr. Trump avoided most direct answers and made just one admission. For the first time, he said he agrees that Russia was behind the cyberattack on the Democratic National Committee and a top aide to campaign rival Mrs. Clinton during the election.

He also tossed in the announcement of his pick to lead the Department of Veterans Affairs, said he would sign executive orders beginning on Jan. 23, and promised to begin negotiating drug prices with pharmaceutical companies to drive costs down.

Questions about his refusal to divest himself of business holdings lingered, though. A few hours after his press conference, U.S. Office of Government Ethics Director Walter Shaub criticized Mr. Trump’s new business arrangement, saying his actions were insufficient to remove potential conflicts.

“Every president in modern times has taken the strong medicine of divestiture,” Mr. Shaub said. “Officials in an administration need their president to show that ethics matter, not only through words but through deeds. This is vitally important if we’re going to have any kind of ethics program.”

On Thursday, Gen. Mattis, testifying before the Senate Armed Services Committee, appeared to buck Mr. Trump numerous times, questioning the motives of Mr. Putin, lauding the North Atlantic Treaty Organization and saying the U.S. should closely monitor Iran’s compliance with a nuclear agreement, but he stopped short of rejecting the deal, as Mr. Trump has.

Gen. Mattis also suggested that some national security discussions could be contentious, which he said would lead to the best outcomes.

“It’s not tidy,” he said of the process he is expecting. “It’ll anticipate that anything but the best ideas will win.”

A day earlier, Rex Tillerson , the pick for secretary of state, had told lawmakers he supported arming Ukraine against Russia and said he was supportive of a trade deal Mr. Obama struck with Asian countries, two statements that conflict with Mr. Trump’s platform.

Later that night, House Speaker Paul Ryan (R., Wis.) said during a CNN town hall that he was working closely with the president-elect to repeal the health-care law but shot down the idea that there would be a “deportation force” to remove illegal immigrants from the U.S. Mr. Trump had said during the campaign that there would be such a force.

Later in the week, Mr. Trump weighed in on the latest development of the issue that dominated the end of the campaign.

He has spent weeks trying to deflect criticism about his election victory, as Democrats argued that Mrs. Clinton had been sandbagged by the Federal Bureau of Investigation’s handling of a probe into whether her private email server had been hacked and whether classified material was improperly moved on it.

The FBI ultimately brought no charges, and on Thursday, the Justice Department’s inspector general confirmed it had opened an investigation into decisions by FBI Director James Comey to make public, days before the election, that agents were scouring a new batch of emails for possible examples of misdeeds by Mrs. Clinton while she was at the State Department. Such a revelation shortly before an election was very unusual.

Mr. Trump on Friday tweeted that the FBI was “VERY nice to her,” adding she “should never…have been allowed to run – guilty as hell.”

In another matter, Mr. Trump during Friday’s interview described a special council, made up of 15 to 20 builders and engineers, that would monitor spending on his $1 trillion plan to improve the nation’s roads, bridges and other public works.

“Some of the projects they’ll throw out, some of the projects they’ll expand, but all of the projects they’ll make sure we get a tremendous bang for the buck,” Mr. Trump said.

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